What does the phrase, a person (who) is “a pair of khaki pants inside a Manila envelope” mean?

I think other answers have missed a couple of things

someone who wears “a pair of khaki pants inside a manila envelope”

Firstly notice that no-one wears pants inside an envelope - that would be impossible. The statement says that Biden is pants inside an envelope.

Secondly, the metaphor says that Biden (a) looks boring from the outside and (b) his personality (his inside) is also boring.

Manila envelope

Khaki pants

In the light of comments I thought I would take the investigation a step further

Perhaps the phrase is more than just a metaphor. I notice that the colours of the Biden/Harris campaign seem to include khaki. (I'm British so I didn't experience this first-hand)

Here's some campaign merchandise

... and here's a pure guess. Could Biden promotional material have arrived in a manila envelope containing a picture of Joe wearing khaki? I don't know. He certainly does wear khaki pants and even suits sometimes.

I'm pretty sure this isn't a "a common, well-received phrase", rather it is a one-off mixed metaphor. I believe it is combining two contexts where a brownish color implies "plain or neutral, not gaudy". Khaki pants or chinos would be a conservative, not-flashy fashion choice in some circles, and a Manila envelope is a plain brown envelope rather than fancy stationery.

I don't believe there is any connotation of "in upper social hierarchy".

Khaki pants are a rather plain, boring type of pants, and a manila envelope is a rather plain, boring envelope. So, I think "a pair of khaki pants inside a manila envelope" is a humorous way of saying "very plain and very boring."

So, Colbert is saying that if Joe Biden turns out to be a very plain and very boring president, he would like that a lot. This would be by contrast with Donald Trump, whose presidency has been anything but plain and boring.